If 0 < α < π/2, then 0 < 2α < π, and the only place between 0 and π where the cotangent equals 0 is at π/2. So
[tex]\begin{aligned}
2\alpha &= \frac {\pi}{2} \\
\alpha &= \frac{\pi}{4}
\end{aligned}[/tex]
... which is the angle you wanted.